Jayanta Basu - cp's OEIS frontend

A228119 Numbers n such that n * (product of digits of n) + 1 is prime.

1, 2, 4, 6, 16, 21, 22, 23, 24, 25, 26, 27, 28, 29, 32, 34, 42, 52, 61, 65, 66, 69, 72, 83, 84, 86, 92, 98, 114, 123, 134, 138, 161, 165, 169, 176, 186, 192, 213, 216, 219, 222, 223, 228, 239, 249, 253, 256, 258, 259, 265, 273, 286, 289, 292, 293, 294, 297

Offset: 1

Jayanta Basu, Aug 14 2013

			16 is a member since 16 * (product of digits of 16) + 1 = 16*6 + 1 = 97 is a prime.

Jayanta Basu, Table of n, a(n) for n = 1..10000

Cf. A000040, A007954.

Select[Range[300], PrimeQ[(#*Times @@ IntegerDigits[#]) + 1] &]

A228096 Numbers consisting of only odd digits such that no permutation of its digits yields a prime.

Original entry on oeis.org

1, 9, 15, 33, 39, 51, 55, 57, 75, 77, 93, 99, 111, 117, 135, 153, 155, 159, 171, 177, 195, 315, 333, 339, 351, 355, 357, 375, 393, 399, 513, 515, 519, 531, 535, 537, 551, 553, 555, 559, 573, 579, 591, 595, 597, 711, 717, 735, 753, 759, 771, 777, 795

Offset: 1

Jayanta Basu, Aug 10 2013

Apart from the first term, A061810 is a subsequence. Conjecture: a(n) ~ A061810(n). - Charles R Greathouse IV, Feb 15 2017

			51 is a member since it consists of only odd digits and both 15 and 51 are composites.

Jayanta Basu, Table of n, a(n) for n = 1..1000

Subsequence of A067013.

Cf. A067013, A061810.

Select[Range[800], And @@ OddQ[x = IntegerDigits[#]] && Count[FromDigits /@ Permutations[x], _?PrimeQ] == 0 &]
Table[FromDigits/@Select[Tuples[Range[1,9,2],n],NoneTrue[FromDigits/@ Permutations[#],PrimeQ]&],{n,3}]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 09 2019 *)

A228102 Numbers n such that sum of all primes <=n is prime.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 10, 13, 14, 15, 16, 37, 38, 39, 40, 43, 44, 45, 46, 281, 282, 311, 312, 503, 504, 505, 506, 507, 508, 541, 542, 543, 544, 545, 546, 557, 558, 559, 560, 561, 562, 593, 594, 595, 596, 597, 598, 619, 620, 621, 622, 623, 624, 625

Offset: 1

Jayanta Basu, Aug 10 2013

nonn

Numbers n such that A034387(n) is prime.

			8 is in the sequence since 2+3+5+7=17 is prime.

Jayanta Basu, Table of n, a(n) for n = 1..1000

Cf. A034387.

[n: n in [1..700] | IsPrime(s) where s is &+PrimesUpTo(n)]; // Bruno Berselli, Aug 10 2013

t = {}; s = 0; Do[If[PrimeQ[n], s += n]; If[PrimeQ[s], AppendTo[t, n]], {n, 625}]; t
Position[Accumulate[Table[If[PrimeQ[n],n,0],{n,700}]],?PrimeQ]//Flatten (* _Harvey P. Dale, Apr 02 2024 *)

A228075 Numbers n whose 10's complement is prime, i.e., 10^k-n, where k is the number of digits of n, is prime.

Original entry on oeis.org

3, 5, 7, 8, 11, 17, 21, 27, 29, 33, 39, 41, 47, 53, 57, 59, 63, 69, 71, 77, 81, 83, 87, 89, 93, 95, 97, 98, 113, 117, 119, 123, 137, 141, 143, 147, 161, 171, 173, 177, 179, 189, 191, 203, 213, 227, 231, 239, 243, 249, 257, 261, 267, 273, 281, 291, 299

Offset: 1

Jayanta Basu, Aug 09 2013

A068811 is a subset.

			8 is a term since 10^1 - 8 = 2 is a prime.
Similarly, 39 is a term as 10^2 - 39 = 61 is prime.

Jayanta Basu, Table of n, a(n) for n = 1..1000

Cf. A068811.

Select[Range[300], PrimeQ[10^(IntegerLength[#]) - #] &]

A225793 Numbers n that can be uniquely expressed as (m + sum of digits of m) for some m.

Original entry on oeis.org

2, 4, 6, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77

Offset: 1

Jayanta Basu, Jul 27 2013

Subset of A176995; first member in A176995 that is not here is 101, next is 103 (cf. A230094).

A230093(a(n)) = 1. - Reinhard Zumkeller, Oct 11 2013

			100 is a member as 100 = 86 + sum of digits of (86). 101 is not a member since both 91 and 100 generate 101. Again 103 is not a member as 92 and 101 generate 103.

Joshi, V. S. A note on self-numbers. Volume dedicated to the memory of V. Ramaswami Aiyar. Math. Student 39 (1971), 327--328 (1972). MR0330032 (48 #8371)
Makowski, Andrzej. On Kaprekar's "junction numbers''. Math. Student 34 1966 77 (1967). MR0223292 (36 #6340)
Narasinga Rao, A. On a technique for obtaining numbers with a multiplicity of generators. Math. Student 34 1966 79--84 (1967). MR0229573 (37 #5147)

Jayanta Basu and Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 1000 terms from Jayanta Basu)
Index entries for Colombian or self numbers and related sequences

Cf. A007953, A062028, A004207, A230093, A003052, A176995, A230094.

a225793 n = a225793_list !! (n-1)
a225793_list = filter ((== 1) . a230093) [1..]
-- Reinhard Zumkeller, Oct 11 2013

For Maple code see A230093. - N. J. A. Sloane, Oct 11 2013

co[n_] := Count[Range[n - 1], _?(# + Total[IntegerDigits[#]] == n &)]; Select[Range[100], co[#] == 1 &]
Select[Tally[Table[m+Total[IntegerDigits[m]],{m,100}]],#[[2]]==1&][[All, 1]]// Sort (* Harvey P. Dale, Aug 23 2017 *)

A227510 Numbers such that product of digits of n is positive and a substring of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51, 61, 71, 81, 91, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 126, 131, 141, 151, 153, 161, 171, 181, 191, 211, 236, 243, 311, 315, 324, 362, 411, 511, 611, 612

Offset: 1

Jayanta Basu, Jul 14 2013

All numbers with at least one zero digit have a product of digits which is a substring; these have been kept out by the restriction on positivity.
The sequence is infinite: if n is a term 10n+1 is also a term. Are there any other patterns (except for prepending 1 to any term)? - Zak Seidov, Jul 24 2013
You can also insert 1 in any position outside the substring that gives the product of digits. - Robert Israel, Aug 26 2014
See also A203566 for a nontrivial subsequence of A203565. The zeroless members of the latter differ from this sequence from 212 on which is there but not here, while 236 is the first here but not there. - M. F. Hasler, Oct 14 2014

			The product of the digits of 236 is 36, a substring of 236, and hence 236 is a member.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A052018, A203565, A203566, A203569, A198298, A236402, A236403, A236404.

filter:= proc(n)
  local L;
  L:= convert(n,base,10);
  if has(L,0) then return false fi;
  verify(convert(convert(L,`*`),base,10),L,'sublist');
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 26 2014

Select[Range[650], FreeQ[x = IntegerDigits[#], 0] && MemberQ[FromDigits /@ Partition[x, IntegerLength[y = Times @@ x], 1], y] &]

{isok(n)=d=digits(n);p=prod(i=1,#d,d[i]);k=1;while(p&&k<=(#d-#digits(p)+1),v=[];for(j=k,k+#digits(p)-1,v=concat(v,d[j]));if(v==digits(p),return(1));k++);return(0);}
n=1;while(n<10^4,if(isok(n),print1(n,", "));n++) \\ Derek Orr, Aug 26 2014

is_A227510(n)={(t=digits(prod(i=1,#n=digits(n),n[i])))&&for(i=0,#n-#t,vecextract(n,2^(i+#t)-2^i)==t&&return(1))} \\ M. F. Hasler, Oct 14 2014

from operator import mul
from functools import reduce
A227510 = [int(n) for n in (str(x) for x in range(1, 10**5)) if not n.count('0') and str(reduce(mul, (int(d) for d in n))) in n]
# Chai Wah Wu, Aug 26 2014

Edited by M. F. Hasler, Oct 14 2014

A227508 Numbers n such that n! is divisible by R(n), the digit reversal of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 18, 19, 20, 21, 22, 23, 24, 25, 27, 29, 30, 31, 32, 33, 36, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Jayanta Basu, Jul 14 2013

Numbers n for which A004086(n) divides A000142(n).

			12 is a member since R(12)=21 divides 12! while 13 is not a member as 31 does not divide 13!.

Cf. A000142, A004086.

read("transforms"):
isA227508 := proc(n)
    if modp(n!,digrev(n)) = 0 then
        true;
    else
        false ;
    end if;
end proc:
A227508 := proc(n)
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA227508(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jul 20 2013

Select[Range[79], Divisible[#!, FromDigits[Reverse[IntegerDigits[#]]]] &]

A226732 Palindromes that cannot be expressed as a nontrivial product of two palindromes.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 101, 111, 131, 141, 151, 161, 171, 181, 191, 212, 232, 252, 272, 292, 313, 323, 343, 353, 373, 383, 414, 434, 454, 474, 494, 515, 525, 535, 545, 565, 575, 585, 595, 656, 676, 717, 727, 737, 747, 757, 767, 787, 797, 818, 838, 858, 878, 898

Offset: 1

Jayanta Basu, Jun 16 2013

Complement of A078337.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A078337.

palQ[n_] := Reverse[x = IntegerDigits[n]] == x; t1 = Rest[Select[Range[900], palQ[#] &]]; Join[{1}, Complement[t1, Union[Flatten[Table[i*j, {i, t1}, {j, t1}]]]]]

A221220 Numbers with more than one prime factor such that concatenation of its prime factors (without multiplicity) is a prime.

Original entry on oeis.org

6, 12, 18, 21, 22, 24, 33, 36, 39, 44, 46, 48, 51, 54, 58, 63, 66, 70, 72, 82, 88, 92, 93, 96, 99, 108, 111, 115, 116, 117, 132, 133, 140, 141, 142, 144, 147, 153, 154, 159, 162, 164, 165, 166, 176, 177, 182, 184, 187, 189, 192, 198, 201, 205, 210, 216, 219

Offset: 1

Jayanta Basu, Jun 04 2013

			Prime factors of 140 are 2, 5, and 7 and 257 is prime, so 140 is a term.

Cf. A024619, A046411.

Select[Range[220],Length[x=First/@FactorInteger[#]]>1&&PrimeQ[FromDigits[Flatten[IntegerDigits[x]]]]&]

from sympy import isprime, primefactors
def ok(n):
    pf = primefactors(n)
    if len(pf) < 2: return False
    return isprime(int("".join(str(p) for p in pf)))
print(list(filter(ok, range(2, 220)))) # Michael S. Branicky, Jun 12 2021

A226218 Ending primes for n-th composite number in the iterated procedure of composite added to sum of prime factors.

Original entry on oeis.org

23, 11, 23, 23, 17, 19, 23, 23, 47, 41, 29, 31, 47, 47, 47, 41, 71, 71, 71, 83, 47, 53, 47, 71, 59, 71, 71, 83, 59, 167, 71, 59, 149, 167, 71, 167, 83, 71, 167, 79, 89, 251, 167, 149, 149, 83, 269, 89, 167, 251, 251, 113, 239, 149, 167, 109, 127, 269, 251, 107

Offset: 1

Jayanta Basu, May 31 2013

nonn

If we consider nonprimes instead of composite then a(1)=2. Sorted list of primes generated here are given in A050778.

			For the first composite number 4 repeated application of composite added to sum of prime factors give 4, 8, 14, 23 and so a(1)=23.

Cf. A050703-A050710, A050778.

a[n_] := NestWhile[#+Total[Times@@@FactorInteger[#]]&, n, !PrimeQ[#]&]; t={}; Do[If[!PrimeQ[n], AppendTo[t, a[n]]], {n, 4, 80}]; t

cp's OEIS Frontend

User: Jayanta Basu

A228119 Numbers n such that n * (product of digits of n) + 1 is prime.

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A228096 Numbers consisting of only odd digits such that no permutation of its digits yields a prime.

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A228102 Numbers n such that sum of all primes <=n is prime.

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A228075 Numbers n whose 10's complement is prime, i.e., 10^k-n, where k is the number of digits of n, is prime.

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A225793 Numbers n that can be uniquely expressed as (m + sum of digits of m) for some m.

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A227510 Numbers such that product of digits of n is positive and a substring of n.

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A227508 Numbers n such that n! is divisible by R(n), the digit reversal of n.

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